stoichiometric analysis of cellular reaction systems

VL Netzwerke, WS 2007/08 Edda Klipp 1

Max Planck Institute Molecular Genetics

Humboldt University BerlinTheoretical Biophysics

Networks in Metabolism and Signaling

Edda Klipp Humboldt University Berlin

Lecture 5 / WS 2007/08Stoichiometry in Metabolic Networks




Stoichiometric Analysis of Cellular Reaction Systems

2A B + C 2D

E F

G

v1 v3

v2

- Analysis of a system of biochemical reactions- Network properties- Enzyme kinetics not considered

http://www.genome.ad.jp/kegg/pathway/map/map01100.html




Stoichiometry and Graphs

http://www.genome.ad.jp/kegg/pathway/map/map01100.html

Enzymes

sMetabolite

E

V

We consider a graph, e.g. a tuple (V,E) with V a set of n vertices and a set of m edges E:

G=(V,E)

Hypergraph




Stoichiometric Coefficients

Stoichiometric coefficients denote the proportions,with which the molecules of substrates and productsenter the biochemical reactions.

Example Catalase2222 OOH2OH2

Stoichiometric coefficients for Hydrogenperoxid, water, oxygen -2 2 1

Stoichiometric coefficients can be chosen such that they agree with molecularity, but not necessarily.

2222 OOHOH2

1

-1 1 1/2

2222 OOH2OH2

2 -2 -1

Their signs depend on the chosen reaction direction. Since reactions are usually reversible,one cannot distinguish between „substrate“ and „product“. v - v




Time Course of Concentrations

Usually described by ordinary differential equations (ODE)

d

d

H O

tv2 2 2 d

d

H O

tv2 2

d

d

O

tv2 Example catalase

for this choice of stoichiometric coefficienten:

2222 OOH2OH2

-2 2 1




Time Course of Concentrations

all rate equations must beconsidered at the same time.

Several reactions at the same time

S 1 S 2

S 3

1 2 3

4

43

322

4211

vS

vvS

vvvS

Usually described by ordinary differential equations (ODE)




Balance equations/Systems equations

In general: We consider the substances Si and their

stoichiometric coefficients nij in the respective reaction j.

If the biochemical reactions are the only reason for the change of concentration of metabolites, i.e. if there is no mass flow by convection, diffusion or similarThen one can express the temporal behavior of concentrations by the balance equations.

d

d

S

tn vi

ij jj

r

1

r – number of reactionSi – metabolite concentrationvj – reaction ratenij – stoichiometric coefficient




The Stoichiometric Matrix

One can summarize the stoichiometric coefficients in matrix N. The rows refer to the substances, the columns refer to the reactions:

Example

S1 S2

S3

1 2 3

4

4321

S

S

S

1000

0110

1011

3

2

1

N

Column: reaction

Row: Substance

External metabolite are not included in N.




Summary

Stoichiometric matrix

Vector of metabolite concentrations

Vector of reaction rates

Parameter vector

ijnN ni ...1 rj ...1

T1,..., nSSS

T1 rvv ,...,v

T1,..., mppp

With N can one write systems equations clearly.

d

d

Sv

tN

Metabolite concentrations and reaction rates are dependent on kinetic parameters.

pSS pSvv ,




The Steady State

Reaction systems are frequently considered in steady state,Where metabolite concentrations change do not change with time.This describes an implicite dependency of concentrationsand fluxes on the parameters.

0pS,vN 0dt

dSb.z.w.

The flux in steady state is p,pSvJ




Concept of Steady States

Restriction of modeling to essential aspectsAnalysis of the asymptotical time behavior of dynamic systemes (i.e. The behavior after sufficient long time span).

Asymptotic behavior can be - oscillatory or - chaotic - in many relevant situations will the system reach a steady state.

The conzept of steady state

- important in kinetic modeling - mathematical idealization

Time




Concept of Steady StatesSeparation of time constants fast and slow processes are coupled

fast processes: initial transition period (often) quasi-steady state

slow processes : change of some quantities in a certain period is often neglectable

(Every steady state can be considered as quasi-steady state embedded in a larger non-stationary system).

Biological organisms are characterized by flow of matter and energytime-independent regimes are usually non-equilibrium phenomena Fließgleichgewicht

Mathematically: replace ODE system(for temporal behavior of variables (concentrations and fluxes))

by an algebraic equation system




Example Unbranched pathway

.constSconstS n .,0

S S1 2, S0 S1 S2 Sn

v1 v2 v3

variabel

3

12

2

11

23122

1211

,

0

0

k

vS

k

vS

SkSkdt

dS

Skvdt

dS

Assumption: Linear kinetics

System equations

Matrix formalism

dS1 / dt = v1-v2

dS2 / dt = v2-v3

d S1 1 -1 0 dt S2 0 1 -1

v1

v2

v3

=

S N v.

.=

Steady state 321 vvv

Nv = 0 is usually a non-linear equation system, which cannotbe solved analytically(necessitates knowledge of kinetic().

dSi /dt = 0




The stoichiometric Matrix N

- Characterizes the network of all reactions in the system

- Contains information about possible pathways

vS

Ntd

d




The Kernel Matrix K

In steady state holds 0d

d v

SN

t

Non-trivial solutions exist only if the columns of N are linearly dependent.

rNRang

Mathematically, the linear dependencies can be expressed by a matrix K withthe columns k which each solve

NK 00Nk

K – null space (Kernel) of N

The number of basis vectors of the kernel of N is NRangr-




Calculation of the Kernel matrix

The Kernel matrix K can be calculated with the Gauss‘ EliminationAlgorith for the solution of homogeneous linear equation systems.

0

0

110

111NK

31

21

11

k

k

k

0

0

3121

312111

kk

kkk

0

1

11

3121

k

kk

Example

Alternative: calculate with computer programmesSuch as „NullSpace[matrix]“ in Mathematica.




Admissible Fluxes in Steady State: Examples

110

011N

111

1k 1kK

S0 S1 S2 S3

v1 v2 v3

Unbranched pathway: one independent steady state flux





S0 S1 S2 S3

v1

v2

v4

v3

1110

0111N

0110

1k

1011

2k

21 kkK





Sv2

v1v3

111 N

0

1

1

1k

1

0

1

2k

21 kkK




Representation of Kernel matrix

The Kernel matrix K is not uniquely determined. Every linear combination of columns is also a Possible solution. Matrix multiplication with a regular Matrix Q „from right“ gives another Kernel matrix.

For some applications one needs a simple ("kanonical") representation of the Kernel matrix.

A possible and appropriate choice is

K contains many Zeros.

KQK ˆ

I

KK 0

I – Identity matrix




Informations from Kernel matrix K

-Admissible fluxes in steady state

-Equilibrium reactions

-Unbranched reaction sequences

-Elementary modes




Admissible Fluxes in steady state

With the vectors ki (k1, k2,…) is also every linear combination A possible columns of K.

for example: instead and also

All admissible fluxes in steady state can be written as linear combinationsof vectors ki :

The coefficients i have the respective units, eg. or .

101

011

21

3

2

1

vvv

v

Mol s 1 mol l h1 1

0110

1k

1011

2k

11

01

0110

1011

123 kkk

Sv2

v1v3for

In steady state holds 0d

d v

SN

t




Equilibrium reactions

Case: all elements of a row in K are 0Then: the respective reaction is in every steady state in equilibrium.

00 j

jj

ijji kv

S1 S2

S3

0 1 2

3

0

1

1

1

K

Example




Kernel matrix –Dead ends

S0 S1 S2 S4

S3

v1 v2 v3

v4

1000

0110

1011

N

S1, S2, S3 intern, S0, S4 extern

Necessary and sufficient condition for a „Dead end“:One metabolite has only one entry in the stoichiometric matrix (is only onceSubstrate or product).

Flux in steady state through this reaction must vanish in steady state (J4 = 0).

Model reduction: one can neglect those reactants for steady state analyses.




Unbranched Reaction Steps

S0 S1 S2 S3

S4

v1 v2 v3

v4

0110

1011N

1

0

0

1

,

0

1

1

1

21 kk

1

0

0

1

0

1

1

1

21

4

3

2

1

v

v

v

v

v

32 vv

The basis vectors of nullspace have the same entries for unbranchedreaction sequences.Unbranched reaction sequences can be lumped for further analysis.




Non-negative Flux Vectors

In many biologically relevant situations have fluxes fixed signs.We can define their direction such that

Sometimes is the value of individual ratesfixed.

Both conditions restrict the freedom for the choice of Basis vectors for K.

0jv

Example: opposite uni-directional rates instead of net rates, - Description of tracer kinetics or dynamics of NMR labels- different isoenzymes for different directions of reactions- for (quasi) irreversible reactions

fixedvk




Kernel Matrix – Irreversibility

S0 S1 S2 S3

S4

v1 v2 v3

v4

0110

1011N

1

0

0

1

,

0

1

1

1

21 kk

2211 kk v02 v

1

1

1

0

,

0

1

1

1

kk02 v Mathematically possible,

biologically not feasible

Other choice of basis vectors

The basis vectors of a null space are not unique.The direction of fluxes (signs) do not necessarily agree with the direction of irreversible reactions.

(Irreversibility limits the space of possible steady state fluxes.)

S1, S2 internal, S0 , S3, S4 external




Elementary Flux Modes

Situation: some fluxes have fixed signes, others can operate in both directions.Which (simple) pathes connect external substrats?

SP1

P2

P3

v1

v2

v3

SP1

P2

P3

v1

v3

111N

v2

1

0

1

,

0

1

1

ik

1

1

0

,

1

0

1

,

0

1

1

,

1

1

0

,

1

0

1

,

0

1

1

ik





-An elementary flux mode comprises all reaction steps, Leading from a substrate S to a product P.

-Each of these steps in necessary to maintain a steady state.

-The directions of fluxes in elementary modes fulfill the demands for irreversibility





S2

S1

S3

P2

P1

P3




Number of elementary flux modes

S0 S1 S2 S3

S4

v1 v2 v3

v4

1

1

1

0

,

1

0

0

1

,

0

1

1

1

,

1

1

1

0

,

1

0

0

1

,

0

1

1

1

k

S0 S1 S2 S3

S4

v1 v2 v3

v4

1

1

1

0

,

1

0

0

1

,

1

0

0

1

,

0

1

1

1

k

The number of elementary modes is at least as high as the number of basisvectors of the null space.




Flux Modes and Extreme Pathways

j

ijji kv

vi

vjvk

NK=0

Extreme pathways:All reactions are irreversible

Flux cone

0jv




Conservation relations: Matrix G

If compounds or groups are not added to or deprived of a Reaction system, then must their total amount remain constant.

0

dt

EESd .constEES Michaelis-Menten kinetics

2 3A BIsolated reaction: .constBA 23

Pyruvatkinase, Na/K-ATPase ATP ADP ADP ATP const .

Examples




Conservation relations - calculation

If there exist linear dependencies between the rows of the stoichiometric matrix, then one can find a matrix G such as

0GN N – stoichiometric matrix

Due tod

d

Sv

tN holds 0

d

d v

SGNG

t

The integration of this equation yields the conservation relations.

.S constG




Conservation relations – Properties of G

The number of independent row vectors g (= number of Independent conservation relations) is given by

Nrankn

(n = number of rows of the stoichiometric matrix = number of metabolites)

GT is the Kernel matrix of NT, and can be found in the same way as K. (Gaussian elimination algorithm)

The matrix G is not unique, with P regular quadratic matrix is again conservation matrix.

PGG

IGG 0Separated conservation conditions:




Conservation relations – Examples

ATP ADPATP

ADP

11

11N

0GN

11G ADP ATP const .

.S constG





Conservation of atoms or atom groups, e.g. Pyruvatdecarboxylase (EC 4.1.1.1)

CH COCOO H CH CHO CO3 3 2

1

1

1

1

N

0

2

1

4

1

2

1

0

0

3

3

3

Gcarbonoxygen

hydrogen

1

0

0

0

1

1

0

1

0

1

0

1

GCH3CO-groupProtons

Carboxyl group

0011 g Elektric charge





v1 v2 v3Glucose Gluc-6P Fruc-6P Fruc-1,6P2

ATP ADP ATP ADP

T

TN

111100

000110

110011

3

2

1

001111

110000

100112

g

g

g

G

232110232 3214 gggg




Conservation Relations – Simplification of the ODE system

If conservation relations hold for a reaction system, then the ODE system can be reduced, since some equations are linearly dependent.

vS N

'N

NN

000 N

L

INLN

'Rearrange N, L – Linkmatrix(independent upper rows, dependent lower rows)

v'S

S

d

dS

d

d 0

b

a NL

I

tt

Rearrange S respectively(indep upper rows, dep lower rows)

d daS t N0 vReduced ODE system

d

d

d

db aS S

t tL' S S const .b a L'For dependent concentrations hold

stoichiometric analysis of cellular reaction systems

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